19 problems

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the Koszul cohomo…

Let $X$ be a smooth projective variety, let $B$ be a line bundle on $X$, let $L_d$ be the asymptotic sequence of line bundles considered in the paper, and define the Betti numbers…

Ein–Lazarsfeld's conjecture. These inequalities are equalities whenever $d\geq b+q+1$. This conjecture concerns the precise first and last nonvanishing positions of Veronese syzygi…

The Geometric Syzygy Conjecture in Even Genus. The last linear syzygy space

Let $X$ be a $g$-dimensional abelian variety over the complex numbers, and let $L$ be an ample line bundle on $X$. For an abelian subvariety $V\subseteq X$, write…

Precise nonvanishing range conjecture. In the situation of that theorem, one has

Asymptotic vanishing conjecture. Fix $q\in[2,n]$. In the situation of the nonvanishing theorem referenced in the source, there is a constant $C_3$, depending on $X$, $A$, $B$ and…

Let $X$ be a smooth projective variety of dimension $n$, let $B$ be a fixed line bundle on $X$, and let $L_d$ be the asymptotically positive line bundles considered above. Write…

Let $R$ be a finitely generated graded $\mathbf{k}$-algebra, let $b$ be fixed, and set $n:=\dim R-1$. For $d>0$, write $R(b)^{(d)}$ for the $d$-th Veronese construction with twist…

Let $K_{p,q}(n,b;d)$ be the Koszul cohomology groups associated with the $d$-th Veronese of $\boldsymbol{\rm P}^n$ and the twist $b$, and consider the vanishing range for these gro…

Let $K_{p,q}$ denote the Koszul cohomology groups governing the Betti table of a high-degree Veronese embedding, and let a row mean the collection of groups with fixed $q$. A Koszu…

Let $X$ be a smooth projective variety of dimension $n$, let $L_d=dA+E$ with $A$ ample and $E$ an arbitrary divisor on $X$, and let $B$ be a line bundle on $X$. For the Koszul coho…

Farkas's conjecture. The curve $C$ has a pure resolution, equivalently it satisfies property $N_p$.

Let $\mathbf{P}^n$ be projective space, let $b\geq 0$, and let $K_{p,q}(\mathbf{P}^n,b;d)$ denote the relevant Koszul cohomology group for the $d$-uple Veronese embedding with coef…

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be an ample divisor, let $P$ and $B$ be divisors, and set … For $2\leq q\leq n$, consider the Koszul cohomology gro…

Minimal Resolution Conjecture. Fix integers $g,r,d,p\geq1$ such that $g-d+r\geq0$, and assume

Minimal Syzygy Conjecture. One has

Let $C$ be a general curve of genus $g=2d-2\geq6$, let $l$ be a prime, and let $\eta\in\operatorname{Pic}^0(C)\setminus\{\mathcal O_C\}$ satisfy $\eta^{\otimes l}=\mathcal O_C$. To…

Let $C$ be a general curve of genus $g=2d-2\geq6$, let $\eta\in\operatorname{Pic}^0(C)$ be a general line bundle, and let $K_{p,q}(C,L)$ denote Koszul cohomology. Green-Lazarsfeld'…